given a function over the quaternions
$$ U(x,y,z,t)+iV(x,y,z,t)+jW(x,y,z,t)+kR(x,y,z,t)=f(x,y,z,t) $$
what are the analogues of the Cauchy Riemann equation for the quaternionic plane so the function defined above is analytic ??
what happens with the Gauss' Theorem ? , so if the function $ f(x,y,z,t) $ is analytic then the integral over a curve in the quaternionic plane is 0 (closed curve)
$$ \oint f(x,y,z,t)ds =0 $$
where is more info about this equation ?? is there a Cauchy's theorem analogue for this integral or Laurent series in the quaternionic plane ??
This question is fairly old but this paper may be of interest to you: https://dougsweetser.github.io/Q/Stuff/pdfs/Quaternionic-analysis-memo.pdf.
It explains exactly what you are looking for, and allow me to paraphrase for simplicity's sake:
Let any quaternion be described as $q= t+ix+jy+kz.$ Allow an analogue to the CR equations for the quaternions to be $$\frac{\partial f}{\partial t}+i\frac{\partial f}{\partial x}+j\frac{\partial f}{\partial y}+k\frac{\partial f}{\partial z}=0.$$
Call any function $f: \mathbb{H} \to \mathbb{H}$ which obeys this equation regular.
Given a regular and continuously differentiable function $f$ and a 3-dimensional manifold on the Quaternions, $C$, the following is true:
$$\int_{C} f(q) \; D_q=0, \\ D_q = (dx \,dy\, dz-i\,dt\, dy \, dz - j\,dt\, dx \, dz - k\,dt \, dx \, dy).$$
I wish there were a simpler definition, but it works just fine.